Conic Sections (Parabola, Ellipse, Hyperbola): JEE Questions & Solutions

Let O be the vertex of the parabola x²=4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ…

Let the foci of a hyperbola coincide with the foci of the ellipse x²/36 + y²/16 = 1. If the eccentricity of the hyperbola is 5, then the…

If the line ax + 4y = √7, where a ∈ ℝ, touches the ellipse 3x² + 4y² = 1 at the point P in the first quadrant, then one of the focal…

Let one end of a focal chord of the parabola y² = 16x be (16, 16). If P(α, β) divides this focal chord internally in the ratio 5: 2, then…

Let y² = 12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that ∠ OPA = 90^°.…

If the chord joining the points P₁(x₁, y₁) and P₂(x₂, y₂) on the parabola y² = 12x subtends a right angle at the vertex of the parabola,…

If the line ax + 2y = 1, where a ∈ ℝ, does not meet the hyperbola x² - 9y² = 9, then a possible value of a is:

Let P(10, 2√15) be a point on the hyperbola x²/a² - y²/b² = 1, whose foci are S and S'. If the length of its latus rectum is 8, then the…

Let the locus of the mid-point of the chord through the origin O of the parabola y² = 4x be the curve S. Let P be any point on S. Then the…

Let S and S' be the foci of the ellipse x²/25 + y²/9 = 1 and P(α,β) be a point on the ellipse in the first quadrant. If (SP)² + (S'P)² - SP…

Let the line y - x = 1 intersect the ellipse x²/2 + y²/1 = 1 at the points A and B. Then the angle made by the line segment AB at the…

Let the domain of f(x) = log₃ log₅ log₇ (9x - x² - 13) be (m, n). Let the hyperbola x²/a² - y²/b² = 1 have eccentricity n/3 and latus…

An equilateral triangle OAB is inscribed in the parabola y² = 4x with the vertex O at the vertex of the parabola. Then the minimum distance…

If the points of intersection of the ellipses x² + 2y² - 6x - 12y + 23 = 0 and 4x² + 2y² - 20x - 12y + 35 = 0 lie on a circle of radius r…

Let PQ be a chord of the hyperbola x²/4 - y²/b² = 1, perpendicular to the x-axis, such that OPQ is an equilateral triangle, O being the…

Let each of the two ellipses E₁:x²/a²+y²/b²=1 (a>b) and E₂:x²/A²+y²/B²=1 (A<B) have eccentricity 4/5. Let the lengths of the latus recta of…

Let the length of the latus rectum of an ellipse x²/a²+y²/b²=1 (a>b) be 30. If its eccentricity is the maximum value of the function…

Let the image of parabola x²=4y in the line x-y=1 be (y+a)²=b(x-c), where a,b,c∈ℕ. Then a+b+c is equal to

Let (h,k) lie on the circle C:x²+y²=4 and the point (2h+1, 3k+2) lie on an ellipse with eccentricity e. Then the value of 5/e² is equal to

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC lying on the line x+2√2 y=4. If the coordinates of the…

For some θ∈(0,π/2), let the eccentricity and the length of the latus rectum of the hyperbola x²-y²sec²θ=8 be e₁ and l₁, respectively, and…

Let the ellipse E: x²/144+y²/169=1 and the hyperbola H: x²/16-y²/λ²=-1 have the same foci. If e and L respectively denote the eccentricity…

An ellipse has its centre at (1,-2), one focus at (3,-2) and one vertex at (5,-2). Then the length of its latus rectum is:

Let A be the focus of the parabola y²=8x. Let the line y=mx+c intersect the parabola at two distinct points B and C. If the centroid of…

Let the parabola y = x² + px - 3 meet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1, -1) passes through…

Let the foci of a hyperbola be (1, 14) and (1, -12). If it passes through the point (1, 6), then the length of its latus-rectum is:

Let P(4, 4√3) be a point on the parabola y² = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars…

Let E: x²/a² + y²/b² = 1, a > b and H: x²/A² - y²/B² = 1. Let the distance between the foci of E and the foci of H be 2√3. If a - A = 2,…

Let A(6,8), B(10cosα, -10sinα), and C(-10sinα, 10cosα) be the vertices of a triangle. If L(a,9) and G(h,k) be its orthocenter and centroid…

If the line 3x - 2y + 12 = 0 intersects the parabola 4y = 3x² at the points A and B, then at the vertex of the parabola, the line segment…

Let the area of a triangle △ PQR with vertices P(5, 4), Q(-2, 4), and R(a, b) be 35 square units. If its orthocenter and centroid are O(2,…

Let the circle C touch the line x - y + 1 = 0, have the center on the positive x-axis, and cut off a chord of length 4/√13 along the line…

A rod of length eight units moves such that its ends A and B always lie on the lines x - y + 2 = 0 and y + 2 = 0, respectively. If the…

Let the shortest distance from (a, 0), where a > 0, to the parabola y² = 4x be 4. Then the equation of the circle passing through the point…

The length of the chord of the ellipse: x²/4 + y²/2 = 1, whose mid-point is ( 1, 1/2), is:

The focus of the parabola y² = 4x + 16 is the center of the circle C with radius 5. If the values of λ, for which C passes through the…

Let the product of the focal distances of the point ( √3, 1/2) on the ellipse x²/a² + y²/b² = 1, (a > b), be 7/4. Then the absolute…

If the equation of the parabola with vertex ( 3/2, 3) and the directrix x + 2y = 0 is α x² + β y² - γ xy - 30x - 60y + 225 = 0, then α + β…

The equation of the chord of the ellipse x²/25 + y²/16 = 1, whose mid-point is (3, 1) is:

Let H₁: x²/a² - y²/b² = 1 and H₂: -x²/A² + y²/B² = 1 be two hyperbolas having lengths of latus rectums 15√2 and 12√5 respectively. Let…

Let ABCD be a trapezium whose vertices lie on the parabola y² = 4x. Let the sides AD and BC of the trapezium be parallel to the y-axis. If…

If A and B are the points of intersection of the circle x² + y² - 8x = 0 and the hyperbola x²/9 - y²/4 = 1, and a point P moves on the line…

If the midpoint of a chord of the ellipse x²/9 + y²/4 = 1 is ( √2, 4/3), and the length of the chord is 2√α/3, then α is:

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y² = 4x with respect to the line x…

Let the ellipse E₁: x²/a² + y²/b² = 1, a > b and E₂: x²/A² + y²/B² = 1, A < B, have the same eccentricity 1/√3. Let the product of their…

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at…

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB,…

If α x + β y = 109 is the equation of the chord of the ellipse x²/9 + y²/4 = 1 whose midpoint is (5/2, 1/2), then α + β is equal to:

Let y² = 12x be the parabola and S its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ) = 147/4. Let C be the circle…

Let one focus of the hyperbola H: x²/a² - y²/b² = 1 be at (√10, 0) and the corresponding directrix be x = 9/√10. If e and l are…

If S and S' are the foci of the ellipse x²/18 + y²/9 = 1 and P be a point on the ellipse, then min(SP · S'P) + max(SP · S'P) is equal to:

Let the focal chord PQ of the parabola y² = 4x make an angle of 60^° with the positive x-axis, where P lies in the first quadrant. If the…

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse…

Let the point P of the focal chord PQ of the parabola y² = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio…

Let A(4, -2), B(1, 1) and C(9, -3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices…

A line passing through the point P(√5, √5) intersects the ellipse x²/36 + y²/25 = 1 at A and B such that (PA).(PB) is maximum. Then 5(PA² +…

The radius of the smallest circle which touches the parabolas y = x² + 2 and x = y² + 2 is

Let the product of the focal distances of the point P(4, 2√3) on the hyperbola H: x²/a² - y²/b² = 1 be 32. Let the length of the conjugate…

Let C be the circle of minimum area enclosing the ellipse E: x²/a² + y²/b² = 1 with eccentricity 1/2 and foci (± 2, 0). Let PQR be a…

The shortest distance between the curves y² = 8x and x² + y² + 12y + 35 = 0 is:

If the equation of the hyperbola with foci (4, 2) and (8, 2) is 3x² - y² - α x + β y + γ = 0, then α + β + γ is equal to

Let the three sides of a triangle are on the lines 4x - 7y + 10 = 0, x + y = 5, and 7x + 4y = 15. Then the distance of its orthocenter from…

The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, -3) and eccentricity is 4/5, is

Let C be the circle x² + (y - 1)² = 2, E₁ and E₂ be two ellipses whose centres lie at the origin and major axes lie on the x-axis and…

The axis of a parabola is the line y = x and its vertex and focus are in the first quadrant at distances √2 and 2√2 units from the origin,…

A line passing through the point A(-2, 0), touches the parabola P: y² = x - 2 at the point B in the first quadrant. The area of the region…

Let the sum of the focal distances of the point P(4, 3) on the hyperbola x²/a² - y²/b² = 1 be 8√(5/3). If for H, the length of the latus…

Let for two distinct values of p, the lines y = x + p touch the ellipse E: x²/4² + y²/3² = 1 at the points A and B. Let the line y = x…

The center of a circle C is at the center of the ellipse E: x²/a² + y²/b² = 1, where a > b. Let C pass through the foci F₁ and F₂ of E such…

Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose…

Consider the hyperbola x²/a² - y²/b² = 1, having one of its foci at P(-3, 0). If the latus rectum through its other focus subtends a right…

Let A = {(α, β) ∈ ℝ × ℝ: |α - 1| ≤ 4 and |β - 5| ≤ 6} and B = {(α, β) ∈ ℝ × ℝ: 16(α - 2)² + 9(β - 6)² ≤ 144}. Then:

Let the length of a latus rectum of an ellipse x²/a² + y²/b² = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t²…

Let e₁ and e₂ be the eccentricities of the ellipse x²/b² + y²/25 = 1 and the hyperbola x²/16 - y²/b² = 1, respectively. If b < 5 and e₁ e₂…

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the…

Let the ellipse 3x² + py² = 4 pass through the centre C of the circle x² + y² - 2x - 4y - 11 = 0 of radius r. Let f₁, f₂ be the focal…

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle α with the positive x-axis and the…

Let r be the radius of the circle, which touches the x-axis at point (a, 0), a < 0 and the parabola y² = 9x at the point (4, 6). Then r is…

If the shortest distance of the parabola y² = 4x from the centre of the circle x² + y² - 4x - 16y + 64 = 0 is d, then d² is equal to:

The length of the chord of the ellipse x²/25 + y²/16 = 1, with midpoint (1, 2/5), is:

If e₁ is the eccentricity of the hyperbola x²/16 - y²/9 = 1 and e₂ is the eccentricity of the ellipse x²/a² + y²/b² = 1, which passes…

Let (5, a/4) be the circumcenter of a triangle with vertices A(a, -2), B(a, 6), and C(a/4, -2). Let α denote the circumradius, β the area,…

If the points of intersection of two distinct conics x² + y² = 4b and x²/16 + y²/b² = 1 lie on the curve y² = 3x², then 3√3 times the area…

Let P(α, β) be a point on the parabola y² = 4x. If P also lies on the chord of the parabola x² = 8y whose midpoint is (1, 5/4), then (α -…

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Let the latus rectum of the hyperbola x²/9 - y²/b² = 1 subtend an angle of π/3 at the center of the hyperbola. If b² is equal to l/m(1+√n),…

Let A(10, 0) and B(0, β) be points on the line 5x + 7y = 50. Let P divide AB in the ratio 7:3. Let 3x-25 = 0 be a directrix of the ellipse…

Let P be a point on the hyperbola H: x²/9 - y²/4 = 1, in the first quadrant such that the area of the triangle formed by P and the two foci…

The area of the region enclosed by the parabola (y-2)² = x-1, the line x-2y+4 = 0, and the positive coordinate axes is:

If the foci of a hyperbola are the same as that of the ellipse x²/9 + y²/25 = 1 and the eccentricity of the hyperbola is 15/8 times the…

Let the foci and length of the latus rectum of an ellipse x²/a² + y²/b² = 1, a > b be (± 5, 0) and √50, respectively. Then, the square of…

Let P be a parabola with vertex (2, 3) and directrix 2x + y = 6. Let an ellipse E with eccentricity 1/√2 pass through the focus of P. The…

Let P be a point on the ellipse x²/9 + y²/4 = 1. Let the line passing through P and parallel to the y-axis meet the circle x² + y² = 9 at…

The vertices of a triangle are A(-1, 3), B(-2, 2), and C(3, -1). A new triangle is formed by shifting the sides of the triangle one unit…

Let C be the circle of minimum area touching the parabola y = 6 - x² and the lines y = √3|x|. Then, which one of the following points lies…

The parabola y² = 4x divides the area of the circle x² + y² = 5 in two parts. The area of the smaller part is:

The area (in square units) of the region enclosed by the ellipse x² + 3y² = 18 in the first quadrant below the line y = x is:

Let the foci of a hyperbola H coincide with the foci of the ellipse E: (x-1)²/100 + (y-1)²/75 = 1, and the eccentricity of the hyperbola H…

Two vertices of a triangle ABC are A(3,-1) and B(-2, 3), and its orthocenter is P(1, 1). If the coordinates of the point C are (α, β) and…

The equations of the sides AB and AC of a triangle ABC are: (λ + 1)x + λ y = 4 and λ x + (1 - λ)y + λ = 0, respectively. Its vertex A is on…

The distance of the point (6, -2√2) from the common tangent y = mx + c, m > 0, of the curves x = 2y² and x = 1 + y² is:

The vertices of a hyperbola H are (±6, 0) and its eccentricity is √5/2. Let N be the normal to H at a point in the first quadrant and…

The equations of two sides of a variable triangle are x = 0 and y = 3, and its third side is a tangent to the parabola y² = 6x. The locus…

Let T and C respectively be the transverse and conjugate axes of the hyperbola 16x² - y² + 64x+ 4y + 44 = 0. Then the area of the region…

If the tangent at a point P on the parabola y² = 3x is parallel to the line x + 2y = 1, and the tangents at the points Q and R on the…

A triangle is formed by the tangents at the point (2, 2) on the curves y² = 2x and x² + y² = 4x, and the line x + y + 2 = 0. If r is the…

If P(h,k) be a point on the parabola x = 4y², which is nearest to the point Q(0, 33), then the distance of P from the directrix of the…

Let A be a point on the x-axis. Common tangents are drawn from A to the curves x² + y² = 8 and y² = 16x. If one of these tangents touches…

Let y = f(x) represent a parabola with focus ( -1/2, 0) and directrix y = -1/2. Then S = { x ∈ ℝ: tan⁻¹ ( √f(x)) + sin⁻¹ ( √(f(x) + 1)) =…

The set of all values of a² for which the line x + y = 0 bisects two distinct chords drawn from a point P( (1 + a)/2, (1 - a)/2) on the…

Let H be the hyperbola, whose foci are (1 ± √2, 0) and eccentricity is √2. Then the length of its latus rectum is:

Let S be the set of all a ∈ ℕ such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈ ℕ on the parabola y² =…

Let P(x₀, y₀) be the point on the hyperbola 3x² - 4y² = 36, which is nearest to the line 3x + 2y = 1. Then √2 (y₀ - x₀) is equal to:

The line x = 8 is the directrix of the ellipse E: x²/a² + y²/b² = 1 with the corresponding focus (2, 0). If the tangent to E at the point P…

If the x-intercept of a focal chord of the parabola y² = 8x + 4y + 4 is 3, then the length of this chord is equal to.

Let the tangents to the curve x² + 2x - 4y + 9 = 0 at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and…

In a group of 100 persons, 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of…

Let the eccentricity of an ellipse x²/a² + y²/b² = 1 be reciprocal to that of the hyperbola 2x² - 2y² = 1. If the ellipse intersects the…

Let R be the focus of the parabola y² = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the point G(10, 10)…

Let the ellipse E: x² + 9y² = 9 intersect the positive x- and y-axes at the points A and B respectively. Let the major axis of E be a…

Let a common tangent to the curves y² = 4x and (x - 4)² + y² = 16 touch the curves at the points P and Q. Then (PQ)² is equal to:

Let a circle of radius 4 be concentric to the ellipse 15x² + 19y² = 285. Then the common tangents are inclined to the minor axis of the…

Consider ellipse E_k: k x² + k² y² = 1, for k = 1, 2, …, 20. Let C_k be the circle which touches the four chords joining the end points…

Let Hₙ: x²/(1 + n) - y²/(3 + n) = 1, n ∈ ℕ. Let k be the smallest even value of n such that the eccentricity of H_k is a rational number.…

If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x² + 4y² = 36 is r, then 12r² is equal to:

Let the tangent to the parabola y² = 12x at the point (3, α) be perpendicular to the line 2x + 2y = 3. Then the square of the distance of…

Let P( 2√3/√7, 6/√7), Q, R, and S be four points on the ellipse 9x² + 4y² = 36. Let PQ and RS be mutually perpendicular and pass through…

If the point (α, 7√3/3) lies on the curve traced by the mid-points of the line segments of the lines xcosθ + ysinθ = 7, θ ∈ (0, π/2),…

Let PQ be a focal chord of the parabola y² = 36x of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be…

Let the tangent and normal at the point ( 3√3, 1) on the ellipse x²/36 + y²/4 = 1 meet the y-axis at the points A and B respectively. Let…

Let (α, β) be the centroid of the triangle formed by the lines 15x - y = 82, 6x - 5y = -4 and 9x + 4y = 17. Then α + 2β and 2α - β are the…

The foci of a hyperbola are (± 2, 0) and its eccentricity is 3/2. A tangent, perpendicular to the line 2x + 3y - 6 = 0, is drawn at a point…

If (α,β) is the orthocenter of the triangle ABC with vertices A(3,-7), B(-1,2), C(4,5), then 9α-6β+60 is equal to

Let an ellipse with centre (1,0) and latus rectum of length 1/2 have its major axis along the x-axis. If its minor axis subtends an angle…